Vector-valued integrals

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1 (Aprl 21, 2014) Vector-valued ntegrals Paul Garrett garrett/ [Ths document s garrett/m/fun/notes /07e vv ntegrals.pdf] 1. Gelfand-Petts ntegrals and applcatons 2. Proof of exstence of Gelfand-Petts ntegrals 3. Totally bounded sets n topologcal vectorspaces 4. Quas-completeness and convex hulls of compacts 5. Hstorcal notes and references Quas-complete, locally convex topologcal vector spaces V have the useful property that contnuous compactly-supported V -valued functons have ntegrals wth respect to fnte Borel measures. Rather than constructng ntegrals as lmts followng [Bochner 1935], [Brkhoff 1935], et ala, we use the [Gelfand 1936]- [Petts 1938] characterzaton of ntegrals, whch has good functoral propertes and gves a forceful reason for unqueness. The ssue s exstence. An mmedate applcaton s to decsve justfcaton of dfferentaton wth respect to a parameter nsde an ntegral, under mld, easly understood hypotheses. Ths s a specal case of a smple general asserton that Gelfand-Petts ntegrals commute wth contnuous operators, descrbed n the frst secton. Another compellng applcaton of ths ntegraton theory s to holomorphc vector-valued functons, wth well-known applcaton to the resolvents of operators on Hlbert and Banach spaces, as n [Dunford 1938] and [Taylor 1938]. In these sources Louvlle s theorem on bounded entre C-valued functons s nvoked to prove that a bounded operator on a Hlbert or Banach spaces has non-empty spectrum. Another of applcaton of holomorphc and meromorphc vector-valued functons s to generalzed functons, as n [Gelfand-Shlov 1964], studyng holomorphcally parametrzed famles of dstrbutons. Many dstrbutons whch are not classcal functons appear naturally as resdues or analytc contnuatons of famles of classcal functons wth a complex parameter. A good theory of ntegraton allows a natural treatment of convolutons of dstrbutons wth test functons, and related operatons. 1. Gelfand-Petts ntegrals and applcatons Let V be a complex topologcal vectorspace, f a measurable V -valued functon on a measure space. A Gelfand-Petts ntegral of f s a vector I f V so that λ(i f ) = λ f (for all λ V ) If t exsts and s unque, ths vector I f s denoted I f = f In contrast to constructon of ntegrals as lmts of fnte sums, ths defnton gves a property that no reasonable noton of ntegral would lack, wthout askng how the property comes to be. Snce ths property s an rreducble mnmum, ths characterzaton of ntegral s a weak ntegral. Unqueness of the ntegral s mmedate when V separates ponts on V, as t does for locally convex V, by Hahn-Banach. Smlarly, lnearty of f I f follows when V separates ponts. Thus, the ssue s exstence. 1

2 [1] The functons we ntegrate are relatvely nce: compactly-supported and contnuous, on measure spaces wth fnte, postve, Borel measures. In ths stuaton, all the C-valued ntegrals λ f exst for elementary reasons, beng ntegrals of compactly-supported C-valued contnuous functons on a compact set wth respect to a fnte Borel measure. The techncal requrement on the topologcal vectorspace V s that the convex hull of a compact set has compact closure. We show below that quas-completeness and local convexty ental ths property. Thus, for example, Hlbert, Banach, Fréchet, LF-spaces and ther weak duals satsfy the hypothess of the theorem. [1.0.1] Theorem: Let be a locally compact Hausdorff topologcal space wth a fnte, postve, Borel measure. Let V be a locally convex topologcal vectorspace n whch the closure of the convex hull of a compact set s compact. Then contnuous compactly-supported V -valued functons f on have Gelfand- Petts ntegrals. Further, ( ) f meas () closure of convex hull of f() [1.0.2] Remark: The concluson that the ntegral of f les n the closure of a convex hull, s a substtute for the estmate of a C-valued ntegral by the ntegral of ts absolute value. [1.0.3] Corollary: Let T : V W be a contnuous lnear map of locally convex topologcal vectorspaces, where convex hulls of compact sets n V have compact closures. Let f be a contnuous, compactly-supported V -valued functon on a fnte regular measure space. Then the W -valued functon T f has a Gelfand-Petts ntegral, and ( ) T f = T f Proof: To verfy that the left-hand sde of the asserted equalty fulflls the requrements of a Gelfand-Petts ntegral of T f, we must show that ( ) µ left-hand sde = µ (T f) (for all µ W ) Startng wth the left-hand sde, ( ( )) µ T f ( ) = (µ T ) f (assocatvty) = (µ T ) f (µ T V and f s a weak ntegral) = µ (T f) (assocatvty) provng that T ( f) s a weak ntegral of T f. /// [1] We do requre that the ntegral of a V -valued functon be a vector n the space V tself, rather than n a larger space contanng V, such as a double dual V, for example. Some alternatve dscussons of ntegraton allow ntegrals to exst n larger spaces. 2

3 [1.0.4] Example: Dfferentaton under the ntegral can be justfed coherently n terms of Gelfand-Petts ntegrals, n many useful stuatons. For example, consder the Banach space C 1( [a, b] [c, d] ) of realdfferentable functons of two real varables on [a, b] [c, d]. The norm s gven by sup of values and sup of (norm of) dervatve. The values of f C 1( [a, b] [c, d] ) and (the norm) of ts dervatve are unformly contnuous, beng contnuous functons on a compact set. Thus, t (x f(x, t)) s a contnuous C 1 [a, b]- valued functon on [c, d]. By desgn, T = t s a contnuous lnear map from V = C1( [a, b] [c, d] ) to W = C o( [a, b] [c, d] ). By the corollary, b b f(x, t) dx = f(x, t) dx t t a a 2. Proof of exstence of Gelfand-Petts ntegrals Agan, unqueness of Gelfand-Petts ntegrals s clear, f they exst. Thus, the ssue s proof of exstence, by a constructon. Proof: To smplfy, dvde by a constant to make have total measure 1. We may assume that s compact snce the support of f s compact. Let H be the closure of the convex hull of f() n V, compact by hypothess. We wll show that there s an ntegral of f nsde H. For a fnte subset L of V, let V L = {v V : λv = λ f, λ L} And let I L = H V L Snce H s compact and V L s closed, I L s compact. Certanly I L I L = I L L for two fnte subsets L, L of V. Thus, f we prove that all the I L are non-empty, then t wll follow that the ntersecton of all these compact sets I L s non-empty. (Ths s the fnte ntersecton property.) That s, we wll have exstence of the ntegral. To prove that each I L s non-empty for fnte subsets L of V, choose an orderng λ 1,..., λ n of the elements of L. Make a contnuous lnear mappng Λ = Λ L from V to R n by Λ(v) = (λ 1 v,..., λ n v) Snce ths map s contnuous, the mage Λ(f()) s compact n R n. For a fnte set L of functonals, the ntegral y = y L = Λf(x) dx s readly defned by component-wse ntegraton. Suppose that ths pont y s n the convex hull of Λ(f()). Snce Λ L s lnear, y = Λ L v for some v n the convex hull of f(). Then Λ L v = y = (..., λ f(x) dx,...) 3

4 Thus, the pont v les n I L as desred. Grantng that y les n the convex hull of Λ L (f(x)), we are done. To prove that y = y L as above les n the convex hull of Λ L (f()), suppose not. From the lemma below, n a fnte-dmensonal space the convex hull of a compact set s stll compact, wthout havng to take closure. Thus, nvokng also the fnte-dmensonal case of the Hahn-Banach theorem, there would be a lnear functonal η on R n so that ηy > ηz for all z n ths convex hull. That s, lettng y = (y 1,..., y n ), there would be real c 1,..., c n so that for all (z 1,..., z n ) n the convex hull In partcular, for all x c z < c y c λ (f(x)) < Integraton of both sdes of ths over preserves orderng, gvng the absurd c y < c y Thus, y does le n ths convex hull. /// [2.0.1] Lemma: The convex hull of a compact set K n R n s compact. In partcular, we have compactness wthout takng closure. Proof: We frst clam that, for a set E n R n and for any x a pont n the convex hull of E, there are n + 1 ponts x 0, x 1,..., x n n E of whch x s a convex combnaton. By nducton, to prove the clam t suffces to consder a convex combnaton v = c 1 v c N v N of vectors v wth N > n + 1 and show that v s actually a convex combnaton of N 1 of the v. Further, we can suppose wthout loss of generalty that all the coeffcents c are non-zero. Defne a lnear map c y L : R N R n R by L(x 1,..., x N ) ( x v, x ) By dmenson-countng, snce N > n + 1 the kernel of L must be non-trval. Let (x 1,..., x N ) be a non-zero vector n the kernel. Snce c > 0 for every ndex, and snce there are only fntely-many ndces altogether, there s a constant c so that cx c for every ndex, and so that cx o = c o for at least one ndex o. Then v = v 0 = c v c x v = (c cx )v Snce x = 0 ths s stll a convex combnaton, and snce cx o = c o at least one coeffcent has become zero. Ths s the nducton, whch proves the clam. Usng the prevous clam, a pont v n the convex hull of K s actually a convex combnaton c o v o +...+c n v n of n + 1 ponts v o,..., v n of K. Let σ be the compact set (c o,..., c n ) wth 0 c 1 and c = 1. The convex hull of K s the mage of the compact set under the contnuous map σ K n+1 L : (c o,..., c n ) (v o, v 1,..., v n ) c v 4

5 so s compact. Ths proves the lemma, fnshng the proof of the theorem. /// 3. Totally bounded sets n topologcal vectorspaces The pont of ths secton s the last corollary, that convex hulls of compact sets n Fréchet spaces have compact closures. Ths s the key pont for exstence of Gelfand-Petts ntegrals. In preparaton, we revew the relatvely elementary noton of totally bounded subset of a metrc space, as well as the subtler noton of totally bounded subset of a topologcal vectorspace. A subset E of a complete metrc space s totally bounded f, for every ε > 0 there s a coverng of E by fntely-many open balls of radus ε. The property of total boundedness n a metrc space s generally stronger than mere boundedness. It s mmedate that any subset of a totally bounded set s totally bounded. [3.0.1] Proposton: A subset of a complete metrc space has compact closure f and only f t s totally bounded. Proof: Certanly f a set has compact closure then t admts a fnte coverng by open balls of arbtrarly small (postve) radus. On the other hand, suppose that a set E s totally bounded n a complete metrc space. To show that E has compact closure t suffces to show that any sequence {x } n E has a Cauchy subsequence. We choose such a subsequence as follows. Cover E by fntely-many open balls of radus 1. In at least one of these balls there are nfntely-many elements from the sequence. Pck such a ball B 1, and let 1 be the smallest ndex so that x 1 les n ths ball. The set E B 1 s stll totally bounded (and contans nfntely-many elements from the sequence). Cover t by fntely-many open balls of radus 1/2, and choose a ball B 2 wth nfntely-many elements of the sequence lyng n E B 1 B 2. Choose the ndex 2 to be the smallest one so that both 2 > 1 and so that x 2 les nsde E B 1 B 2. Proceedng nductvely, suppose that ndces 1 <... < n have been chosen, and balls B of radus 1/, so that x E B 1 B 2... B Then cover E B 1... B n by fntely-many balls of radus 1/(n+1) and choose one, call t B n+1, contanng nfntely-many elements of the sequence. Let n+1 be the frst ndex so that n+1 > n and so that x n+1 E B 1... B n+1 Then for m < n we have d(x m, x n ) 1 m so ths subsequence s Cauchy. /// In a topologcal vectorspace V, a subset E s totally bounded f, for every neghborhood U of 0 there s a fnte subset F of V so that E F + U Here the notaton F + U means, as usual, F + U = v + U = {v + u : v F, u U} v F 5

6 [3.0.2] Remark: In a topologcal vectorspace whose topology s gven by a translaton-nvarant metrc, a subset s totally bounded n ths topologcal vectorspace sense f and only f t s totally bounded n the metrc space sense, from the defntons. [3.0.3] Lemma: In a topologcal vectorspace the convex hull of a fnte set s compact. Proof: Let the fnte set be F = {x 1,..., x n }. Let σ be the compact set σ = {(c 1,..., c n ) R n : c = 1, 0 c 1, for all } R n Then the convex hull of F s the contnuous mage of σ under the map (c 1,..., c n ) c x so s compact. /// [3.0.4] Proposton: A totally bounded subset E of a locally convex topologcal vectorspace V has totally bounded convex hull. Proof: Let U be a neghborhood of 0 n V. Let U 1 be a convex neghborhood of 0 so that U 1 + U 1 U. Then for some fnte subset F we have E F + U 1, by the total boundedness. Let K be the convex hull of F, whch by the prevous result s compact. Then E K + U 1, and the latter set s convex, as observed earler. Therefore, the convex hull H of E les nsde K + U 1. Snce K s compact, t s totally bounded, so can be covered by a fnte unon Φ + U 1 of translates of U 1. Thus, snce U 1 + U 1 U, H (Φ + U 1 ) + U 1 Φ + U Thus, H les nsde ths fnte unon of translates of U. Ths holds for any open U contanng 0, so H s totally bounded. /// [3.0.5] Corollary: In a Fréchet space, the closure of the convex hull of a compact set s compact. Proof: A compact set n a Fréchet space (or n any complete metrc space) s totally bounded, as recalled above. By the prevous result, the convex hull of a totally bounded set n a Fréchet space (or n any locally convex space) s totally bounded. Thus, ths convex hull has compact closure, snce totally bounded sets n complete metrc spaces have compact closure. /// 4. Quas-completeness and convex hulls of compacts The followng proof borrows an dea from the proof of the Banach-Alaoglu theorem. It reduces the general case to the case of Fréchet spaces, treated n the prevous secton. [4.0.1] Proposton: In a quas-complete locally convex topologcal vectorspace, the closure C of the convex hull H of a compact set K s compact. Proof: Snce s locally convex, by the Hahn-Banach theorem ts topology s gven by a collecton of semnorms v. For each semnorm v, let v be the completon of the quotent /{x : v(x) = 0} wth respect to the metrc that v nduces on the latter quotent. Thus, v s a Banach space. Consder Z = v v (wth product topology) 6

7 wth the natural njecton j : Z, and wth projecton p v to the v th factor. By constructon, and by defnton of the topology gven by the semnorms, j s a (lnear) homeomorphsm to ts mage. That s, s homeomorphc to the subset j of Z, gven the subspace topology. The mage p v jk s compact, beng a contnuous mage of a compact subset of. Snce v s Fréchet, the convex hull H v of p v jk has compact closure C v. The convex hull jh of jk s contaned n the product v H v of the convex hulls H v of the projectons p v jk. By Tychonoff s theorem, the product v C v s compact. Snce jc s contaned n the compact set v C v, to prove that the closure jc of jh n j s compact, t suffces to prove that jc s closed n Z. Snce jc s a subset of the compact set v C v, t s totally bounded and so s certanly bounded (n Z, hence n j). By the quas-completeness, any Cauchy net n jc converges to a pont n jc. Snce any pont n the closure of jc n Z has a Cauchy net n jc convergng to t, jc s closed n Z. Ths fnshes the proof that quas-completeness mples the compactness of closures of compact hulls of compacta. /// 5. Hstorcal notes and references Most nvestgaton and use of ntegraton of vector-valued functons s n the context of Banach-space-valued functons. Nevertheless, the dea of [Gelfand 1936] extended and developed by [Petts 1938] mmedately suggests a vewpont not confned to the Banach-space case. A hnt appears n [Rudn 1991]. Ths s n contrast to many of the more detaled studes and comparsons of varyng notons of ntegral specfc to the Banach-space case, such as [Bochner 1935]. A varety of developmental epsodes and results n the Banach-space-valued case s surveyed n [Hldebrandt 1953]. Proofs and applcaton of many of these results are gven n [Hlle-Phllps 1957]. (The frst edton, authored by Hlle alone, s sparser n ths regard.) See also [Brooks 1969] to understand the vewpont of those tmes. One of the few exceptons to the apparent lmtaton to the Banach-space case s [Phllps 1940]. However, t seems that n the Unted States after the Second World War consderaton of anythng fancer than Banach spaces was not popular. The present pursut of the ssue of quas-completeness (and compactness of the closure of the convex hull of a compact set) was motvated orgnally by the dscusson n [Rudn 1991], although the latter does not make clear that ths condton s fulflled n more than Fréchet spaces, and does not menton quas-completeness. Imagnng that these deas must be applcable to dstrbutons, one mght cast about for means to prove the compactness condton, eventually httng upon the hypothess of quas-completeness n conjuncton wth deas from the proof of the Banach-Alaoglu theorem. Indeed, n [Bourbak 1987] t s shown (by apparently dfferent methods) that quas-completeness mples ths compactness condton, although there the applcaton to vector-valued ntegrals s not mentoned. [Schaeffer-Wolff 1999] s a very readable account of further mportant deas n topologcal vector spaces. The fact that a bounded subset of a countable strct nductve lmt of closed subspaces must actually be a bounded subset of one of the subspaces, easy to prove once conceved, s attrbuted to Deudonne and Schwartz n [Horvath 1966]. See also [Bourbak 1987], III.5 for ths result. Pathologcal behavor of uncountable colmts was evdently frst exposed n [Douady 1963]. Evdently quotents of quas-complete spaces (by closed subspaces, of course) may fal to be quas-complete: see [Bourbak 1987], IV.63 exercse 10 for a constructon. [Brkhoff 1935] G. Brkhoff, Integraton of functons wth values n a Banach space, Trans. AMS 38 (1935), [Bochner 1935] S. Bochner, Integraton von Funktonen deren Werte de Elemente enes Vektorraumes snd, 7

8 Fund. Math. 20 (1935), Paul Garrett: Vector-valued ntegrals (Aprl 21, 2014) [Bourbak 1987] N. Bourbak, Topologcal Vector Spaces, ch. 1-5, Sprnger-Verlag, [Brooks 1969] J.K. Brooks, Representatons of weak and strong ntegrals n Banach spaces, Proc. Nat. Acad. Sc. U.S., 1969, [Douady 1963] A. Douady, Partes compactes d un espace de fonctons contnues a support compact, C. R. Acad. Sc. Pars 257 (1963), [Dunford 1936] N. Dunford, Integraton of abstract functons, abstract, Bull. AMS 44 (1936), 178. [Dunford 1938] N. Dunford, Unformty n lnear spaces, Trans. AMS 44 (1938), [Gelfand 1936] I. M. Gelfand, Sur un lemme de la theore des espaces lneares, Comm. Inst. Sc. Math. de Kharkoff, no. 4, vol. 13, 1936, [Gelfand-Shlov 1964] I. M. Gelfand, G. E. Shlov Generalzed Functons, vol. 1, translated by E. Saletan, Academc Press, [Hldebrandt 1953] T. H. Hldebrandt, Integraton n abstract spaces, Bull. AMS, 59 (1953), [Hlle-Phllps 1957] E. Hlle wth R. Phllps, Functonal Analyss and Semgroups, AMS Colloquum Publcaton, second edton, [Horvath 1966] J. Horvath, Topologcal Vector Spaces and Dstrbutons, Addson-Wesley, [Petts 1938] B. J. Petts, On ntegraton n vector spaces, Trans. AMS 44 (1938), [Phllps 1940] R. S. Phllps, Integraton n a convex lnear topologcal space, Trans. AMS, 47 (1940), [Rudn 1991] W. Rudn, Functonal Analyss, second edton, McGraw-Hll, [Schaeffer-Wolff 1999] H.H. Schaeffer, wth M.P. Wolff, Topologcal Vector Spaces, second edton, Sprnger, [Taylor 1938] A. E. Taylor, The resolvent of a closed transformaton, Bull. AMS 44 (1938), [Taylor 1971] A. E. Taylor, Notes on the hstory of the uses of analytcty n operator theory, Amer. Math. Monthly 78 (1971),